An important problem in statistics is that of making inference after model selection. We consider here a situation in survival analysis where the model is composed of two parametric families: the Weibull family and the gamma family. Given is a sample X₁, X₂ ,... , X_n from a distribution F, where it is known that its distribution F either belongs to the Weibull or the gamma families, we examine the question of how to estimate a common parameter γ, such as the mean of F. We present two approaches of estimating γ: a nonparametric approach, which ignores the distributional information, and a two-step approach wherein we first decide on the model using the data, and then we estimate the parameter assuming that the chosen model is correct. We show through simulation studies the behavior of the estimators and compare their performances using mean-squared error (MSE).

With repeated measures data, it is important to model an appropriate covariance structure in order to make valid inferences. Three criteria that can be used to select the best covariance structure are the AIC, AICC, and BIC fit statistics. Another possible strategy is to always assume a certain structure, say compound symmetry. A simulation study was conducted to investigate the different strategies in order to determine if there is a best way to select a variance-covariance structure. Data sets were generated for 60 scenarios under the null hypotheses (no time effect and no interaction). The scenarios were created based on combinations of five different variance-covariance structures, two levels of number of groups, two levels of number of time periods, and three sample sizes per group. Each data set was analyzed in SAS PROC MIXED, selecting from five different covariance structures. Analyses were done treating time as both categorical and numeric variables. Each data set was analyzed using three denominator degrees of freedom methods. Results indicate that the AICC and BIC selection strategies are better at maintaining the nominal significance level than the AIC. With very small sample sizes, all of the fit statistic strategies over-estimate the nominal significance level. With moderately small n (n=10 to 15), AICC is generally the better method in selecting a covariance structure. With a total sample size of 20 (k=2, n=10) or more, the BIC performs about the same or better than the AICC. With large sample sizes, all three methods tend to work well. With one group, the AICC performs better than the BIC in both classifications of time, but with two groups the BIC performs as well as or better than the AICC, regardless of the classification of time. Also, the results for the time-by-group interaction follow a pattern similar to the results for time with two groups. The between-within and Satterthwaite denominator degrees of freedom methods produce similar results, and are better than the residual method in maintaining the significance level, especially when treating time as numeric.

A joint model of longitudinal marker and recurrent event is established. This general model accommodates internal (marker) covariate, external covariate, accumulation effect and intervention effect. A finite mixture model is employed to handle underlying heterogeneous population. A likelihood approach is adopted to obtain parameter estimates via EM algorithm. Model selection, seed value of iteration by cluster analysis are considered.

Spline modeling may provide a better fit to data than polynomial or categorical models. There is no one best approach, however, as some modeling methods may produce better results for predicted values (e.g., smaller confidence intervals) than other methods, depending on the data. To address this, a simulation study was undertaken. Data were simulated for five different structures (patterns), using one dependent variable and one independent variable. Each of these five structures was generated with three different sample sizes, and two different standard deviations, for a total of 30 scenarios. Six different regression models were evaluated for each of the 30 scenarios: simple linear regression (SLR), polynomial regression (quadratic and cubic), and spline regression (linear, quadratic and cubic). Selected measures were compared for each scenario to assess model "fit" verses model simplicity: confidence interval coverage on the predicted values, confidence interval widths, mean squared error (MSE), and the PRESS and R² statistics. Results indicated that the best choice of a modeling method should take into account preliminary plots and the estimated standard deviation. Splines are most appropriate when the plots of the data clearly indicate that they are needed (i.e., when the standard deviation is small and we can detect knots and changes in structure). This is especially true of the linear spline, which is clearly most suited for piecewise linear data structures. When the plots do not show much detail (i.e., when the standard deviation is large), it is generally better to use a simpler model (e.g., polynomial). Results also reinforce the need to look at a plot of the predicted values for your model, as some of the usual selection criteria (MSE, PRESS, R²) can give similar results for various models, but the coverage for these models may be diverse.

Iterated Total Time on Test (TTT) transforms can be used to characterize certain stochastic orderings. In particular, orderings with respect to expectation of functions, convex with respect to polynomials, are equivalent to higher order TTT domination under certain conditions. This ordering gives rise to a k-mart structure, which gives meaningful representation for the relationship of two jointly distributed random variables. Here we illustrate these ideas applied to mixtures of gamma distributions.

This talk will focus on techniques to find likelihood estimates of a mixture of two three-parameter gammas via the ECM algorithm and ways to accelerate that algorithm via methods by Louis (1982) and Meilijson (1989).